2.3.3A Time

Grade: 
2
Subject:
Math
Strand:
Geometry & Measurement
Standard 2.3.3

Use time and money in real-world and mathematical situations.

Benchmark: 2.3.3.1 Time

Tell time to the quarter-hour and distinguish between a.m. and p.m.

Overview

Big Ideas and Essential Understandings 

3 students with clocks

Standard 2.3.3 Essential Understandings

Second graders tell time to the quarter-hour and understand the difference between am and pm. They find the value of a group of coins including quarters, dimes, nickels and pennies. They find combinations of coins that equal a given amount.

All Standard Benchmarks

2.3.3.1
Tell time to the quarter-hour and distinguish between a.m. and p.m.
2.3.3.2
Identify pennies, nickels, dimes and quarters.  Find the value of a group of coins and determine combinations of coins that equal a given amount.

Benchmark Cluster 
Benchmark Group A

2.3.3.1
Tell time to the quarter-hour and distinguish between a.m. and p.m.

What students should know and be able to do [at a mastery level] related to these benchmarks:
  • tell time to the quarter-hour on an analog clock.
  • use the language "quarter to," "quarter after'" "fifteen minutes after" and "fifteen minutes before."
  • understand the difference between am and pm.
Work from previous grades that supports this new learning includes: 
  • tell time to the hour and half-hour.
  • visually discriminate between the hour hand and minute hand.
  • understand the concepts of before and after.
  • skip count by 5s and 10s.
Correlations 

NCTM Standards

Understand measurable attributes of objects and the units, systems, and processes of measurement.

Pre-K - 2 Expectations
  • recognize the attributes of length, volume, weight, area, and time;
  • compare and order objects according to these attributes;
  • understand how to measure using nonstandard and standard units;
  • select an appropriate unit and tool for the attribute being measured.

Common Core State Standards

Work with time and money.
  • 2.MD.7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
  • 2.MD.8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Misconceptions

Student Misconceptions 
Student Misconceptions and Common Errors

Students may think...

  • the hour hand is the minute hand and vice versa.
  • the hour numeral is also the minute numeral.

Resources

Instructional Notes 
Teacher Notes
  • Students may need support in further development of previously studied concepts and skills
  • Working with individual clocks with hands in two different colors can help them distinguish the two hands. Label the hands and have children use them in their work. Point out to the children that the word "minute" is "longer" than the word "hour" which is "shorter". Many opportunities to explain what each hand shows will help students.
  • When students are able to tell how many minutes after the hour, introduce reading minutes before the hour.
  • Students may incorrectly name 15-minute intervals on an analog clock. On a demonstration clock, display and name 15-minute intervals of time. Emphasize that there are four quarter-hours on a clock.
  • In learning to tell time, it's important to link time to daily activities. Throughout the school day find opportunities to focus reading the time on a clock.
  • Hang a digital clock next to an analog clock on the wall. At significant times during the day, stop and read the times as students compare the two clocks.
  • Before and after are important terms in learning to order events and measure time. Asking questions, such as "How many minutes before lunch?" will reinforce students' learning of time.
  • To build understanding of the quarter-hour, invite students to fold a paper clock into 4 parts. They should fold the clock in half on the 12 and 6, then in quarters on the 3 and 9. Students could color each quarter of the clock a different color.
  • Questioning

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication.  A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources 
NCTM Illuminations

This lesson provides an introduction to and practice with the concept of time. The activities focus students' attention on the attributes of time and enable students at varying levels to develop knowledge and skills in using time.

Additional Instructional Activities

Students make a bingo card showing different ways to say times on a clock. Write the time choices on the board for students to copy onto their game cards. Students write the times in any box they choose. In this way, each card will have a different sequence of times.

quarter to ten

eight fifteen

half past ten

twelve fifteen

quarter after six

five thirty

four forty-five

two fifteen

seven thirty

ten fifteen

quarter after one

eight forty-five

free

two thirty

one fifteen

three thirty

seven forty-five

quarter past eight

ten thirty

eleven fifteen

five fifteen

noon

six o'clock

two o'clock

eleven o'clock

Begin a bingo game by showing the times written on the bingo cards on a clock. Students read the clock and cover the time on their bingo card. Rules of the game will vary--five in a row horizontally or vertically, four corners, etc.

Students can practice and build their time skills utilizing the many time activities.

Playing the game of "I Have, Who Has" is an engaging way to practice telling time.  Directions and game cards to play can be found at this link.

Additional Instructional Resources    

Dacey, L., Cavanagh, M., Findell, C. R., Greenes, C., Jensen Sheffield, L., & Small, M. (2003). Navigating through measurement in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

New Vocabulary 

a.m. - (Ante Meridiem - Latin for "before midday") The hours from midnight to noon are a.m. hours.

p.m. - (Post Meridiem - Latin for "after  midday") The hours from noon to midnight are p.m. hours.

quarter-hour - a unit of time equal to 15 minutes.

"Vocabulary literally is the

 key tool for thinking."

Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful Use: Multiple and varied opportunities to use the words in context.  These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Word Bank

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Frayer Model

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Example / Non-Example chart

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

word

definition

illustration

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Professional Learning Communities 


Reflection - Critical Questions regarding the teaching and learning of these benchmarks:

What are the key ideas related to telling time at the second grade level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to tell time to the quarter hour successfully?

What common errors do second graders make when telling time?

When checking for student understanding of telling time at the second grade level, what should teachers:

  • listen for in student conversations?
  • look for in student work?
  • ask during classroom discussions?

Examine student work related to a task involving telling time. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the second grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8. (2nd. ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

References 

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, Marilyn. (2007). About teaching mathematics:  A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA:  Houghton Mifflin Company.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.     

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.

Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Assessment

  • Draw the hands to show 2:45. 

Draw hands to show 2:45.

2:45 (solution)Solution:

 

Benchmark:   2.3.3.1

  • Devin eats dinner at quarter past six.  Which clock shows quarter past six?

A.clock A

B.clock B

    clock C

C.

D.clock D


Images from http://etc.usf.edu//clipart/sitemap/clocks.php
Solution:          C.
Benchmark:     2.3.3.1

 

  • What time does the clock show?  _________________

What time does the clock show?


Solution:          1:15
Benchmark:     2.3.3.1

  • At which time are you most likely to go swimming?

A.   2:00 a.m.
B.   2:00 p.m.
C.   6:00 a.m.
D. 10:00 p.m.

Solution:          B. 2:00 p.m.
Benchmark:     2.3.3.1

  • What time does the clock show?

What time is it? (9:30)

A.  6:45
B.  9:06
C.  9:30
D.  6:15

Solution:          C. 9:30
Benchmark:     2.3.3.1

  • True or False:

You are likely to eat lunch at 12:00 a.m.

Solution:          False
Benchmark:     2.3.3.1

  • Write the time in two ways.  (11:30, 30 minutes after 11 o'clock) (DOK Level 2)

11:30

 ____ minutes after ____ o'clock

Differentiation

Struggling Learners 

To help students connect time to their daily routine and experiences, they could make a book describing their school day. Have them stamp a clock face on each page and write a time given by the teacher. When that time of day occurs, ask them to draw the hands on the clock to match the time. They can draw a picture to show what they are doing in school at that time of day.

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

              Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations. 

The CRA Trinagle

Additional Resources

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2 Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

English Language Learners 

Language for telling time can be especially confusing. Words like quarter-past, quarter-to will need to be explained and illustrated during instruction. Students will need to use clocks with movable hands as they construct the concept of telling time.  Connecting times on the clock to daily activities will help students understand the concept of time.

If students are familiar with Spanish words, help them relate the English words.  The following words are cognates:  hour (hora), quarter-hour (cuarto de hora), minute (minuto), and seconds (segundos).  Say the English words and have them say them with you.

  • Word banks need to be part of the student learning environment in every mathematics unit of study.  Refer to these throughout instruction.
  • Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as quarter hour a.m., p.m..

Frayer Model

  • Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

A quarter past means _________________________________________________.

The time is _______________ because ____________________________________.

  • When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.
Additional ELL Resources

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 

Students that are successful at telling time, may enjoy investigating How old are you in... weeks?  hours?  minutes?  seconds?

They may be interested in investigating

The History of Telling Time

or U.S. Time Zones.

 

Additional Resources

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Parents/Admin

Classroom Observation 
Administrative/Peer Classroom Observation

Students are ...

Teachers are...

distinguishing between a.m. and p.m.

explaining the meaning of a.m. (Ante Meridiem - before noon - the midpoint of the day) and p.m. (post meridian - after noon - the midpoint of the day).

telling time to the quarter-hour.

helping students understand the meaning of quarter hour, quarter-to and quarter-past.

demonstrating their understanding of time by showing various times on small hand-held clocks.

infusing the use of time throughout the day. For example, helping students note the start time or end times of daily activities.

Additional Resources

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

For Administrators

Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Parents 

Parent Resources

Mathematics handbooks to be used as home references:

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Helping your child learn mathematics  Provides activities for children in preschool through grade 5

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

Help Your Children Make Sense of Math


Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995